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## Homework Statement

Let A,B ε Mnxn(C)

Prove that if B is invertible, then there exists a scalar cεC s.t. A + cB is not invertible. HINT:

examine det (A+cB)

## Homework Equations

## The Attempt at a Solution

I know that the det(A+cB)=0 since a non invertible matrix has det=0

I know B is invertible so I multiply the right side by B^-1 which gives det(AB^-1 + cIn)=0 where In is the nxn identity matrix. There is a theorem that says that a scalar is an eigenvalue of A if and only if det(A-cIn)=0.

can I choose c to be -c? does that show that A +cB is not invertible? Help!